12 Standard Notes
12 Standard Notes
12 Standard Notes
Let A and B be .two non-empty sets, then a function f from set A to set B is a rule
whichassociates each element of A to a unique element of B.
1. Algebraic Functions
Some algebraic functions are given below
(i) Polynomial Functions If a function y = f(x) is given by
Polynomial Functions
where, a0, a1, a2,…, an are real numbers and n is any non -negative integer, then f
(x) is called a polynomial function in x.
is a polynomial of degree 5.
(iii) Irrational Functions The algebraic functions containing one or more terms
having nonintegral rational power x are called irrational functions.
e.g., y = f(x) = 2√x –3√x + 6
2. Transcendental Function
A. function, which is not algebraic, is called a transcendental function.
Trigonometric, Inverse trigonometric, Exponential, Logarithmic, etc are
transcendental functions.
Intervals of a Function
(i) The set of real numbers x, such that a ≤ x ≤ b is called a closed interval and
denoted by [a, b] i.e., {x: x ∈ R, a ≤ x ≤ b}.
(ii) Set of real number x, such that a < x < b is called open interval and is
denoted by (a, b) i.e., {x: x ∈ R, a < x < b}
Domain of f{x) = R
Range of f{x) = {c}
2. Identity Function
The function that associates to each real number x for the same number x, is called
the identity function. i.e., y = f(x) = x, ∀ x ∈
R. Domain of f(x) = R
Range f(x) = R
Identity Function
3. Linear Function
If a and b be fixed real numbers, then the linear function is defmed as y = f(x) =
ax + b, where a and b are constants.
Domain of f(x) = R
Range of f(x) = R
Linear Function
4. Quadratic Function
If a, b and c are fixed real numbers, then the quadratic function is expressed as y
= f(x) = ax2 + bx + c, a ≠ 0 ⇒ y = a (x + b / 2a)2 + 4ac – b2 / 4a
which is equation of a parabola in downward, if a < 0 and upward, if a > 0 and
vertex at ( – b / 2a, 4ac – b2 / 4a).
Domain of f(x) = R
Range of f(x) is [ – ∞, 4ac – b2 / 4a], if a < 0 and [4ac – b2 / 4a, ∞], if a > 0
5. Square Root Function Square root function is defined by y = F(x) = √x, x ≥ 0.
Quadratic Function
6. Exponential Function
Exponential function is given by y = f(x) = ax, where a > 0, a ≠ 1.
Exponential Function
7. Logarithmic Function
A logarithmic function may be given by y = f(x) = loga x, where a > 0, a ≠ 1 and x
> 0.
The graph of the function is as shown below. which is increasing, if a > 1 and
decreasing, if 0 < a < 1.
Logarithmic Function
8. Power Function
The power function is given by y = f(x) = xn ,n ∈ I,n≠ 1, 0. The domain and range
of the graph y = f(x), is depend on n.
Power Function
Domain of f(x) = R
Range of f(x) = [0, ∞)
Power Function
Domain of f(x) = R
Range of f(x) = R
(c) If n is negative even integer.
Power Function
Modulus Function
Domain of f(x) = R
Range of f(x) = [0, &infi;)
signum function
Domain of f(x) = R
Range of f(x) = {-1, 0, 1}
where, [x] represents the greatest integer less than or equal to x. i.e., for any
integer n, [x] = n, if n ≤ x < n + 1 Domain of f(x) = R Range of f(x) = I
∴ f(x) = (x)
Domain of f = R
Range of f= [x] + 1
(i) Domain = R
(ii) Range = [-1,1]
(iii) Period = 2π
2. Graph of cos x
Graph of cos x
(i) Domain = R
(ii) Range = [-1,1]
(iii) Period = 2π
3.Graph of tan x
Graph of tan x
4. Graph of cot x
Graph of cot x
5. Graph of sec x
Graph of sec x
6. Graph of cosec x
Graph of cosec x
(i) Domain = R ~ nπ, n ∈ I
(ii) Range = [- &infi;, – 1] ∪ [1, &infi;)
(iii) Period = 2π
(i) Sum The sum of the functions f and g is defined as f + g : X → R such that (f +
g) (x) = f(x) + g(x).
(ii) Product The product of the functions f and g is defined as fg : X → R, such
that (fg) (x) = f(x) g(x) Clearly, f + g and fg are defined only, if f and g have
the same domain. In case, the domain of f and g are different. Then, Domain of f +
g or fg = Domain of f ∩ Domain of g.
(iii) Multiplication by a Number Let f : X → R be a function and let e be a real
number .
Then, we define cf: X → R, such that (cf) (x) = cf (x), ∀ x ∈ X.
(iv) Composition (Function of Function) Let f : A → B and g : B → C be two
functions. We define gof : A → C, such that got (c) = g(f(x)), ∀ x ∈ A
Alternate There exists Y ∈ B, such that if f(x) = y and g(y) = z, then got (x) = z
Periodic Functions
A function f(x) is said to be a periodic function of x, provided there exists a
real number T > 0, such that F(T + x) = f(x), ∀ x ∈ R
The smallest positive real number T, satisfying the above condition is known as the
period or the fundamental period of f(x) ..
Testing the Periodicity of a Function
(i) Put f(T + x) = f(x) and solve this equation to find the positive values of T
independent of x.
(ii) If no positive value of T independent of x is obtained, then f(x) is a non-
periodic function.
(iii) If positive val~es ofT independent of x are obtained, then f(x) is a periodic
function and the least positive value of T is the period of the function f(x).
Inverse of a Function
Let f : A → B is a bijective function, i.e., it is one-one and onto function.
Inverse of a Function
PART 2
Relations and Functions Math Notes Download PDF
Important Links
Chapter 5: Continuity and Differentiability
Chapter 1: Relations and Functions
Chapter 2: Inverse Trigonometric Functions
Chapter 3: Matrices
Chapter 4: Determinants
Chapter 6: Application of Derivatives
Chapter 7: Integrals
Chapter 8: Application of Integrals
Chapter 9. Differential Equations
Chapter 10: Vector Algebra
Chapter 12: Linear Programming
Chapter 11: Three Dimensional Geometry
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